# Algebraic Bethe Ansatz for spinor R-matrices

### Submission summary

 As Contributors: Vidas Regelskis Preprint link: scipost_202108_00062v2 Date submitted: 2021-11-01 19:34 Submitted by: Regelskis, Vidas Submitted to: SciPost Physics Academic field: Physics Specialties: Mathematical Physics Approach: Theoretical

### Abstract

We present a supermatrix realisation of $q$-deformed spinor-spinor and spinor-vector $R$-matrices. These $R$-matrices are then used to construct transfer matrices for $U_{q^2}(\mathfrak{so}_{2n+1})$- and $U_{q}(\mathfrak{so}_{2n+2})$-symmetric closed spin chains. Their eigenvectors and eigenvalues are computed.

###### Current status:
Editor-in-charge assigned

I would like to thank the referees for carefully reading the manuscript and for their useful comments and suggestions. I have corrected the typos and implemented the suggested improvements.

### List of changes

The changes made are listed in the replies to the referee reports.

### Submission & Refereeing History

Resubmission scipost_202108_00062v2 on 1 November 2021
Submission scipost_202108_00062v1 on 25 August 2021

## Reports on this Submission

### Report

I do think that the new version is good for publication.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Report

After author's corrections I think that the paper can be published in SciPost Physics.

• validity: good
• significance: good
• originality: high
• clarity: good
• formatting: good
• grammar: good

### Vidas Regelskis  on 2021-11-02  [id 1903]

Dear Editor,

I would like to thank the referees for carefully reading the manuscript and for their useful comments and suggestions. I have corrected the typos and implemented the suggested improvements.

Response to Referee 1

I thank the referee for raising the point about the possible contradiction between the commutation relations of fundamental $L$-operators and their coproduct rule. Recall that the universal $R$-matrix $\mathcal{R}$ is an element in a completion of $U_q(\mathfrak{b}_+) \otimes U_q(\mathfrak{b}_-)$ where $\mathfrak{b}_\pm$ are the standard Borel subalgebras. There is a number of ways of defining the $L$-operators consistently. For instance,

$L^+(u) = (\pi_u \otimes \text{id})\, \mathcal{R}^{-1} , \qquad L^-(u) = (\text{id} \otimes \pi_u)\, \mathcal{R} , \qquad R(u,v) = (\pi_u \otimes \pi_v)\, \mathcal{R}^{-1}$

gives the wanted defining relations and the wanted coproduct rule,

$\Delta( L^\pm_{ij}(u) ) = \sum_{k} L^\pm_{ik}(u) \otimes L^\pm_{kj}(u).$

For $U_q(\widehat{\mathfrak{gl}}_N)$ this was explicitly demonstrated by J. Ding and I. B. Frenkel in Comm. Math. Phys. 156, 277-300 (1993) and by E. Frenkel and E. Mukhin in Sel. Math. 8, 537-635 (2002). For $U_q(\widehat{\mathfrak{so}}_N)$ this could be verified using the isomorphism constructed recently by N. Jing, M. Liu and A. Molev in SIGMA 16, 043, (2020).

Response to Referee 2

I thank the referee for raising the point about way the twist matrices are moved through the $R$-matrices in the proof of Theorem 3.3 and Theorem 4.4. This was indeed overlooked in the first version paper. This issue has been fixed. The twist matrices are now included in the definition of the monodromy matrices, given by eqs. (3.2) and (4.3). (Tables on pages 19 and 26 were updated accordingly.) This is a valid construction since the twist matrices satisfy the fundamental exchange relations, (2.65) and (2.70). This yields the wanted results without needing to deal with the twist matrices explicitly.

Response to Referee 3

I thank the referee for the points they have raised. Below I list my responses and the changes I have made:

1.1. The deformation parameter of $U_{q^2}(\mathfrak{so}_{2n+1})$ is set to $q^2$ to avoid having $\sqrt{q}$ in the spinor-spinor $R$-matrix and the corresponding exchange relations. The square root of the deformation parameter arises because the root system of $\mathfrak{so}_{2n+1}$ has a simple short root. (This is not the case for $\mathfrak{so}_{2n+2}$ since its simple roots are all of the same length.) This explanation was added to the Introduction (page 2, line 45) and Section 2.5 (page 8, line 176).

2.1. It is indeed possible to obtain the vector-vector $R$-matrix by fusing spinor-vector $R$-matrices. However this construction is not needed for the goals of this paper, hence is not included.

3.1. The $q$-transposition defined by (2.6-2.7) and all instances of it were renamed to a new symbol.

3.2. The ambiguous notation of the creation operators was fixed. The repeated symbol $\beta$ was replaced by $\mathscr{b}$.

3.3. The notation below line 140 on page 6 is correct. Here $x_j^{m_j}$ with $m_j = 0, 1$ denote elements of the exterior algebra $\Lambda$ defined in line 138. In particular, $x_j^0 = 1$ and $x_j^1=x_j$.

3.4. The notation in equation (2.30) is correct. Here $\omega_i$ is an element of the deformed Clifford algebra $\mathscr{C}^n_q$.

4.1. I have added an explanation of the product notation below line 109 on page 5.

4.2. The ambiguous notation $(\dot a \ddot a)^j_n$ was replaced by $a^j_n$. An explanation of this notation was added at the beginning of Section 3.1 on page 18 and at the beginning of Section 4.1 on page 25.

4.3. The algebras $U_{q^2}(\mathfrak{so}_{2n+1})$ and $U_q(\mathfrak{so}_{2n})$ are explicitly mentioned in the Abstract. I have not included them in the title to avoid having mathematical symbols and thus help the search engines to index the paper.

Kind regards, Vidas Regelskis